Mean Value Theorem

f ' (c)= [f(b)-f(a)]/ (b-a)

So the Mean Value Theorem states that if you are given a function that is continuous and differentiable (which just means no jumps, holes, or corners) the slope of the secant line and the tangent line are the same at point "c". This means that they are two parallel lines.

Alright, so let's try this out. We will be using a very familiar graph to make things a litttle easier to see and understand. (At least i worked for me)

We have the graph f(x)= x² on [0,2]

So a=0 and b=2. Let's get this numbers in the formula [f (b)-f (a)]/(b-a)

1) [f(2)-f(0)]/(2-0)= 4-0/2=2 so m=2

2) Now we will replace x by the two we just got into the f(x)>>>>f(2)=4

Now we have a point and a slope so let's use the point-slope formula

3) y-4= 2(x-2) which is y=2x. This is our secant equation. (The green line on the graph)

We know the f '(x) = 2x so f '(c) =2c

We also know that 2c= slope and we know the slope is 2. Therefore c=1.

Now we find out what f(c) equals when c=1>>>>>> its 1 so we now know the tangent line has to go throught the point (1,1) with the slope 2x. We do the point-slope formula.

4) y-1= 2(x-1)

y-1=2x-2

y= 2x-1 (It's the red line)

So why doesnt this work on a function that's not smooth? Let me show you with an example AKA another graph.

As you can see the tangent line doesn't exist at x=0

What is f '(x) saying about f(x)

This is the graph f '(x) which simply means that what you see on this graph are the slopes of the graph f(x). So what is this graph telling us about our invisible f(x).

1) Where is the functions f(x) increasing? Where is it decreasing? And how exactly do we know?

Ok, so as I mentioned before, this is the graph of the slope of f(x) so where is the slope increasing? F(x) is increasing from (-2,o) U (0, 2). When f '(x) >0 the function f(x) is increasing. -2,0,2 are exclusive because at those points the slope is zero. F(x) is decreasing from (-∞, 2) U (2, ∞) because when f '(x) <0> the function f(x) is decreasing. I a visual learner so for those like me here you go. Hope it helps.

2) Where is there an extrema?

So, to answer this question you must already know that to find an extrema you have to find either the end points or the critical points. Because this graph has no end points we have to find critical points. A critical point is found when f '(x) is either Zero or Undefined. As you can clearly see on the given graph f '(0) is zero so your extrema is at (0,0).

3) Where does f(x) concave up? Where does it concave down?

F(x) is concave up when the slope of f '(x) is positive which is approximately from (-∞, -1.25) U (0, 1.25) and concaves down approximately from (-1.25, 0) U (1.25, ∞)

4) So what does f(x) look like on a graph?

It has to be a x^5 because f '(x) is x^4 and the anti derivative has to reach x to the power of five.

Stiff Muscle

1. Which mindset do you think you are a part of when it comes to "intelligence"? According to the reading, what tells you that you are of this mindset?

When it comes to intelligence I think there are some things you can learn and some things that take longer for one to understand. Well, when I started reading, I noticed that I leaned more on the fixed than the growing mind set when it came to math. I have always felt that my math skills aren't that great. For this reason, I sometimes have my doubts and ask myself why did I even take this class. I'm usually not a quitter and enjoy the academic challenges that are put upon me, but with math I don't see it as a challenge. To me calculus is beyond what I can do and the only reason I stayed was because my Math Teacher truly believes I am capable of doing this and succeeding.

2. How has this mindset helped or hurt you in math?

Well, it surely hasn't helped me at all. It has actually held me back from what I could probably do. When it comes to math I seem to be easily discourage and I know this isn't the way it should work but I can't seem to help it. I just say "Math is NOT my subject" and stick with that. It's hard to change the habit of saying that. I've said for quite a while I'm beginning to believe the my subconscious believes it.

3. What is your reaction to finding out that the brain is just a big muscle that can be trained?

My dad has told me that before so it wasn't a big surprise. Maybe my brain is just one of those stiff muscles that needs more time to train when it comes to math.

4. How do you see this new piece of information affecting your future?

I guess the more logical answer to this question would be "Yeah I will definitely change and this information will help me in the future" but my problem hasn't been not knowing this information. I guess it would be believing it and putting it to use. I really do want to be better in math but sometimes I get mad at myself because I can't seem to understand the material as fast as other, as fast as I want to understand.

I guess I should go easier on me and believe in myself more than I do...like Ms.Hwang believes I can.